Freshwater Holiday Parc in Burton Bradstock ticks all those boxes, but it is VERY holiday park-ish. Has it’s own section of beach, swimming pools with splashy kids stuff, ten pin bowling, bars etc. Five minutes from Bridport.
There’s a few campsites in Charmouth (great beach with safe swimming- much of Chesil beach isn’t, fossils, slightly crap pubs).
Lyme Regis area has loads.
Have a look at https://coolcamping.com/campsites/uk/england/west-country/dorset
If you want to stay in the heart of ‘Broadchurch’ West Bay, near Bridport, has a few campsites + holiday parks. Again, is busy and touristy.
Basic but blissful: https://www.nationaltrust.org.uk/golden-cap/features/st-gabriels-campsite-on-the-golden-cap-estate